 
Summary: LOCAL RINGS OVER WHICH ALL MODULES
HAVE RATIONAL POINCAR
E SERIES
Luchezar L. Avramov
To Steve Halperin on February 1
Abstract. If the homotopy Lie algebra (R) of a local ring R contains a free Lie subalge
bra of nite codimension, then for each nitely generated R{modules M the Poncare series
P R
M (t) =
P 1
n=0 dim k Tor R
n (M; k) t n represents a rational function in t, and there is a least
common denominator for all these functions. When this denominator is a power of (1 t),
the ring R is a complete intersection, which has at most one nonquadratic dening equation.
Introduction
Let R be a commutative noetherian local ring with maximal ideal mR and residue
eld k, and let M be a nitely generated R{module. An important characteristic of
M is contained in its Betti sequence f b R
n (M) = dim k Tor R
n (M; k) g n0 , which renes the
